Applications
Attitude toward abortion
The dataset gss08 taken from the 2008 General Social
Survey (Smith et al., 2010) includes six binary manifest items
measuring 355 respondents’ attitudes toward abortion following the
strategy suggested by McCutcheon, 1987. For each item, respondents were
asked whether abortion should be legalized under various circumstances:
a strong chance of serious defect in the baby (DEFECT),
pregnancy is seriously endangering the woman’s health
(HLTH), pregnancy as a result of rape (RAPE),
due to low income, the family cannot afford any more children
(POOR), woman is unmarried and has no plans to marry the
man (SINGLE), and woman is married but does not want more
children (NOMORE). Each item has two possible levels of
response (i.e., "YES" or "NO").
gss08 also include five covariates: AGE,
GENDER, RACE, DEGREE and
REGION of respondents. (?gss08)
data("gss08")
str(gss08)
#> 'data.frame': 355 obs. of 11 variables:
#> $ DEFECT: Factor w/ 2 levels "YES","NO": 1 2 2 1 1 1 1 1 1 1 ...
#> $ HLTH : Factor w/ 2 levels "YES","NO": 1 2 2 1 1 1 1 1 1 1 ...
#> $ RAPE : Factor w/ 2 levels "YES","NO": 1 2 2 1 1 1 1 1 1 1 ...
#> $ POOR : Factor w/ 2 levels "YES","NO": 2 2 2 1 1 1 1 1 1 1 ...
#> $ SINGLE: Factor w/ 2 levels "YES","NO": 2 2 2 1 1 1 1 1 NA 1 ...
#> $ NOMORE: Factor w/ 2 levels "YES","NO": 1 2 2 1 1 1 1 NA 1 1 ...
#> $ AGE : num 32 21 56 40 62 71 58 36 52 78 ...
#> $ SEX : Factor w/ 2 levels "MALE","FEMALE": 1 2 2 1 2 1 2 1 1 2 ...
#> $ RACE : Factor w/ 3 levels "WHITE","BLACK",..: 2 2 2 2 1 2 2 1 1 1 ...
#> $ DEGREE: Factor w/ 4 levels "<= HS","HIGH SCHOOL",..: 2 2 1 4 3 2 3 3 2 3 ...
#> $ REGION: Factor w/ 9 levels "NEW ENGLAND",..: 2 2 2 2 2 2 2 2 2 2 ...Selecting the number of latent classes: In the
first step of the analysis, we conduct a series of standard latent class
models to select a number of latent classes. The number of class in the
glca() function is set to 2, 3, or 4 as following commands.
It should be noted that we use 10 sets of randomly generated initial
parameters to avoid the problem of local maxima (i.e.,
n.init = 10), and the argument is set as
seed = 1 in the glca() function to ensure
reproductibility of results unless otherwise noted.
f <- item(DEFECT, HLTH, RAPE, POOR, SINGLE, NOMORE) ~ 1
lca2 <- glca(f, data = gss08, nclass = 2, seed = 1, verbose = FALSE)
lca3 <- glca(f, data = gss08, nclass = 3, seed = 1, verbose = FALSE)
lca4 <- glca(f, data = gss08, nclass = 4, seed = 1, verbose = FALSE)gofglca(lca2, lca3, lca4, test = "boot", seed = 1)
#> Model 1: item(DEFECT, HLTH, RAPE, POOR, SINGLE, NOMORE) ~ 1
#> nclass: 2
#> Model 2: item(DEFECT, HLTH, RAPE, POOR, SINGLE, NOMORE) ~ 1
#> nclass: 3
#> Model 3: item(DEFECT, HLTH, RAPE, POOR, SINGLE, NOMORE) ~ 1
#> nclass: 4
#>
#> Goodness of Fit Table :
#> logLik AIC CAIC BIC entropy Res.Df Gsq Boot p-value
#> 1 -740.10 1506.20 1569.43 1556.43 0.95 50 135.13 0.00
#> 2 -687.45 1414.90 1512.17 1492.17 0.88 43 29.83 0.36
#> 3 -684.19 1422.38 1553.70 1526.70 0.79 36 23.31 0.56
#>
#> Analysis of Deviance Table :
#> npar logLik Df Deviance Boot p-value
#> 1 13 -740.10
#> 2 20 -687.45 7 105.31 0.00
#> 3 27 -684.19 7 6.52 0.26The output from the function gofglca() comprises of two
tables; goodness-of-fit table and analysis-of-deviance table. The former
shows model fit criteria such as AIC, CAIC, BIC, and entropy, \(G^2\) statistic, and its bootstrap \(p\)-value for absolute model fit. In this
example, the bootstrap \(p\)-values
indicate that the two-class model (Model 1) fits data poorly (\(p\)-value = 0.00), but the three-class and
the four-class models (Model 2 and Model 3) fit data adequately (\(p\)-value = 0.36 and 0.56, respectively).
The latter table displays deviance statistic comparing two competing
models and its bootstrap \(p\)-value
for the relative model fit. The null hypothesis in the test for
comparing Model 1 and Model 2 (i.e., the fit of the two-class model is
not significantly poorer than the fit of the three-class model) should
be rejected (\(p\)-value = 0.00). In
the test for comparing Model 2 and Model 3, however, the bootstrap \(p\)-value (= 0.26) indicates that the fit
of the four-class model has not been improved significantly compared to
the fit of the three-class model. In addition, the three-class model has
the smallest value among these three models in the model fit criteria.
Therefore, we can conclude that the three-class model is an appropriate
for the gss08 data.
summary(lca3)
#>
#> Call:
#> glca(formula = f, data = gss08, nclass = 3, seed = 1, verbose = FALSE)
#>
#> Manifest items : DEFECT HLTH RAPE POOR SINGLE NOMORE
#>
#> Categories for manifest items :
#> Y = 1 Y = 2
#> DEFECT YES NO
#> HLTH YES NO
#> RAPE YES NO
#> POOR YES NO
#> SINGLE YES NO
#> NOMORE YES NO
#>
#> Model : Latent class analysis
#>
#> Number of latent classes : 3
#> Number of observations : 352
#> Number of parameters : 20
#>
#> log-likelihood : -687.4486
#> G-squared : 29.82695
#> AIC : 1414.897
#> BIC : 1492.17
#>
#> Marginal prevalences for latent classes :
#> Class 1 Class 2 Class 3
#> 0.34467 0.46396 0.19138
#>
#> Class prevalences by group :
#> Class 1 Class 2 Class 3
#> ALL 0.34467 0.46396 0.19138
#>
#> Item-response probabilities (Y = 1) :
#> DEFECT HLTH RAPE POOR SINGLE NOMORE
#> Class 1 0.8275 0.9453 0.7960 0.0638 0.0390 0.1344
#> Class 2 1.0000 1.0000 1.0000 0.9813 0.9284 0.9657
#> Class 3 0.0466 0.3684 0.0949 0.0000 0.0000 0.0000
#>
#> Item-response probabilities (Y = 2) :
#> DEFECT HLTH RAPE POOR SINGLE NOMORE
#> Class 1 0.1725 0.0547 0.2040 0.9362 0.9610 0.8656
#> Class 2 0.0000 0.0000 0.0000 0.0187 0.0716 0.0343
#> Class 3 0.9534 0.6316 0.9051 1.0000 1.0000 1.0000
plot(lca3)
Considering group variable and testing the measurement
invariance As the group information provided in the data, we
can consider the multilevel data structure and compare the latent class
structure between higher-level units (i.e., groups). The
glca() function can incorporate group variable by setting
group argument as the name of group variable in the data. For example,
in order to investigate whether attitudes toward legalizing abortions
vary by the final degree of respondents, we may set DEGREE
as group variable by typing group = DEGREE in the
glca() function . DEGREE is coded into four
categories (i.e., "<= HS", "HIGH SCHOOL",
"COLLEGE", and "GRADUATE") indicating from
under high school diploma to graduate degree. Moreover, we can implement
the test for measurement invariance across groups using the
glca() function. The measurement invariance assumption can
be adjusted through measure.inv argument in
glca(). The default is measure.inv = TRUE,
constraining item-response probabilities to be equal across groups. The
following commands implement two different tests for group variable: (1)
test whether class prevalence is significantly influenced by group
variable under the measurement-invariant model; and (2) test whether the
measurement models differ across groups.
mglca1 <- glca(f, group = DEGREE, data = gss08, nclass = 3, seed = 1, verbose = FALSE)
mglca2 <- glca(f, group = DEGREE, data = gss08, nclass = 3, measure.inv = FALSE, seed = 1, verbose = FALSE)
gofglca(lca3, mglca1, mglca2, test = "chisq")
#> Model 1: item(DEFECT, HLTH, RAPE, POOR, SINGLE, NOMORE) ~ 1
#> nclass: 3
#> Model 2: item(DEFECT, HLTH, RAPE, POOR, SINGLE, NOMORE) ~ 1
#> Group: DEGREE, nclass: 3, measure.inv: TRUE
#> Model 3: item(DEFECT, HLTH, RAPE, POOR, SINGLE, NOMORE) ~ 1
#> Group: DEGREE, nclass: 3, measure.inv: FALSE
#>
#> Goodness of Fit Table :
#> logLik AIC CAIC BIC entropy Res.Df Gsq
#> 1 -687.45 1414.90 1512.17 1492.17 0.88 43 29.83
#> 2 -672.41 1396.83 1523.28 1497.28 0.88 229 87.85
#> 3 -650.95 1461.89 1850.98 1770.98 0.89 175 44.91
#>
#> Analysis of Deviance Table :
#> npar logLik Df Deviance Pr(>Chi)
#> 1 20 -687.45
#> 2 26 -672.41 6 30.07 0.00
#> 3 80 -650.95 54 42.94 0.86Since the model specified in the object lca3 (Model 1 in
the output from the goflca() function) is constructed with
six binary items, the number of parameters for the saturated models is
\(26 − 1 = 63\). Therefore, degree of
freedom for Model 1 is \(63 − 20 =
43\). The models specified in mglca1 and mglca2 (Model 2 and
Model 3) are involved with group variable with four categories, and the
number of parameters for the saturated models is \(26 \times 4 − 1 = 255\). Therefore, degrees
of freedom for Model 2 and Model 3 are \(255 −
26 = 229\) and \(255 − 80 =
175\), respectively. It should be noted that model comparison has
been conducted through chi-squares by setting
test = "chisq" because Model 1 is nested in Model 2, and
Model 2 is nested in Model 3. The analysis-of-deviance table provided by
the gofglca() function shows that the chi-square \(p\)-value for comparing Model 1 and Model 2
is 0.00, while the \(p\)-value for
comparing Model 2 and Model 3 is 0.86. Hence, we can deduce that the
measurement invariance assumption can be assumed, but class prevalences
vary across levels of DEGREE.
Testing the equality of coefficients across groups:
We can further consider the subject-specific covariates which may
influence the probability of the individual belonging to a specific
class. Covariates such as AGE, RACE, and
SEX in the gss08 dataset can be incorporated
into the model specified in mglca1. AGE is
respondent’s age and considered as a numeric variable in the
glca() function. The respondent’s race, RACE
has three levels (i.e., "WHITE", "BLACK", and
"OTHER"), and the respondent’s gender, SEX is
coded as two categories (i.e., "MALE" and
"FEMALE"). We can easily implement the test for exploring
group differences using the gofglca() function. For
example, the following commands implement two different tests for
SEX: (1) test whether the class prevalence is significantly
influenced by SEX under the model where the coefficients
are constrained to be identical across groups; and (2) test for
assessing group difference in the effect of SEX on class
prevalence by specifying the model where coefficients are allowed to
vary across groups and comparing it to the model where coefficients are
constrained to be identical across groups. Since the iteration is
initiated with random parameters, the order of class labels can be
switched under the identical ML solution. Therefore, the random seed is
set as seed = 3 in the glca() function for the
model specified in mglcr1 below to ensure identical class
order as provided in Figure 1.
fx <- item(DEFECT, HLTH, RAPE, POOR, SINGLE, NOMORE) ~ SEX
mglcr1 <- glca(fx, group = DEGREE, data = gss08, nclass = 3, seed = 3, verbose = FALSE)
mglcr2 <- glca(fx, group = DEGREE, data = gss08, nclass = 3, coeff.inv = FALSE, seed = 1, verbose = FALSE)
gofglca(mglca1, mglcr1, mglcr2, test = "chisq")
#> Model 1: item(DEFECT, HLTH, RAPE, POOR, SINGLE, NOMORE) ~ 1
#> Group: DEGREE, nclass: 3, measure.inv: TRUE
#> Model 2: item(DEFECT, HLTH, RAPE, POOR, SINGLE, NOMORE) ~ SEX
#> Group: DEGREE, nclass: 3, measure.inv: TRUE, coeff.inv: TRUE
#> Model 3: item(DEFECT, HLTH, RAPE, POOR, SINGLE, NOMORE) ~ SEX
#> Group: DEGREE, nclass: 3, measure.inv: TRUE, coeff.inv: FALSE
#>
#> Goodness of Fit Table :
#> logLik AIC CAIC BIC entropy Res.Df Gsq
#> 1 -672.41 1396.83 1523.28 1497.28 0.88 229 87.85
#> 2 -666.71 1389.42 1525.60 1497.60 0.88 323 149.97
#> 3 -662.04 1392.09 1557.45 1523.45 0.88 317 140.64
#>
#> Analysis of Deviance Table :
#> npar logLik Df Deviance Pr(>Chi)
#> 1 26 -672.41
#> 2 28 -666.71 2 11.41 0.00
#> 3 34 -662.04 6 9.33 0.16The models specified in mglcr1 and mglcr2
(Model 2 and Model 3 in the output from the goflca()
function) are involved with an additional covariate, SEX, and the number
of possible cases is \(26 \times 4 \times 2 −
1 = 511\). However, as only 352 observations are used for the
analysis, the number of parameters for the saturated model becomes \(352 − 1 = 351\). Therefore, degrees of
freedom for Model 2 and Model 3 are \(351 − 28
= 323\) and \(351 − 34 = 317\),
respectively. The analysis-of-deviance table provided by the
gofglca() function shows that SEX has a significant impact
on the class prevalence (\(p\)-value =
0.00) when we compare Model 1 with Model 2. However, the model without
any constraint on coefficients (Model 3) is not significantly superior
to Model 2 (\(p\)-value = 0.16),
indicating that the impact of SEX is not group specific.
Note that Model 2 is mathematically equivalent to the standard LCR with
covariates DEGREE and SEX without the
interaction terms, but Model 2 is more intuitive and useful
configuration when comparison of latent structures by group is a major
concern.
Summarizing the results from the selected model: Based on the previous analysis, we can conclude that measurement models are equivalent across groups (i.e., measurement invariance assumption is satisfied) in the three-class latent class model. In addition, there is a significant effect of SEX on the class prevalence, but there is no group difference in the amount of effect.
Figure 1 displays the estimated parameters from the selected model
specified in mglcr1 using the command
plot(mglcr1). The line graph in Figure 1 displays the
estimated item-response probabilities. We can see that the identified
three classes are clearly distinguished by item-response probabilities.
Class 1 represents individuals who are in favor of all the six reasons
for abortion, while Class 3 represents those who consistently oppose all
the six reasons. Individuals in Class 2 seem to distinguish between
favoring the first three reasons (i.e., DEFECT,
HLTH, and RAPE) and opposing the last three
reasons (i.e., POOR, SINGLE, and
NOMORE) for abortion. The first bar graph in Figure 1
describes the estimated marginal class prevalences, and the second bar
graph displays the estimated class prevalences for each category of
group variable. The values in parentheses printed in the
x-axis are the group prevalences. Figure 1 displays that
the class prevalence varies across groups, and the respondents with
higher degrees are more advocate for legalizing abortion than those with
lower degrees. The covariate effect can be confirmed through the Wald
test for each of estimated odds ratios and coefficient by the command
coef(mglcr1).
The SEX coefficient results the same, while the
intercept determining the class prevalence differs across groups in the
selected model. To avoid redundancy and save space, only the results of
the SEX coefficient are displayed as follows, but the
entire output of the object can be displayed by the command
summary(mglcr1) in the R console.
coef(mglcr1)
#> Coefficients :
#>
#> Class 1 / 3 :
#> Odds Ratio Coefficient Std. Error t value Pr(>|t|)
#> SEXFEMALE 0.32824 -1.11400 0.09062 -12.29 <2e-16 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> Class 2 / 3 :
#> Odds Ratio Coefficient Std. Error t value Pr(>|t|)
#> SEXFEMALE 0.35678 -1.03064 0.09971 -10.34 <2e-16 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
summary(mglcr1)
#>
#> Call:
#> glca(formula = fx, group = DEGREE, data = gss08, nclass = 3,
#> seed = 3, verbose = FALSE)
#>
#> Manifest items : DEFECT HLTH RAPE POOR SINGLE NOMORE
#> Grouping variable : DEGREE
#> Covariates (Level 1) : SEX
#>
#> Categories for manifest items :
#> Y = 1 Y = 2
#> DEFECT YES NO
#> HLTH YES NO
#> RAPE YES NO
#> POOR YES NO
#> SINGLE YES NO
#> NOMORE YES NO
#>
#> Model : Multiple-group latent class analysis
#>
#> Number of latent classes : 3
#> Number of groups : 4
#> Number of observations : 352
#> Number of parameters : 28
#>
#> log-likelihood : -666.7097
#> G-squared : 149.9656
#> AIC : 1389.419
#> BIC : 1497.601
#>
#> Marginal prevalences for latent classes :
#> Class 1 Class 2 Class 3
#> 0.46144 0.33996 0.19860
#>
#> Class prevalences by group :
#> Class 1 Class 2 Class 3
#> <= HS 0.16848 0.51010 0.32143
#> HIGH SCHOOL 0.44386 0.34339 0.21275
#> COLLEGE 0.55347 0.29616 0.15036
#> GRADUATE 0.71183 0.20237 0.08580
#>
#> Logistic regression coefficients :
#> Group : <= HS
#> Class 1/3 Class 2/3
#> (Intercept) 0.0811 1.1445
#> SEXFEMALE -1.1140 -1.0306
#>
#> Group : HIGH SCHOOL
#> Class 1/3 Class 2/3
#> (Intercept) 1.424 1.1261
#> SEXFEMALE -1.114 -1.0306
#>
#> Group : COLLEGE
#> Class 1/3 Class 2/3
#> (Intercept) 2.0173 1.3487
#> SEXFEMALE -1.1140 -1.0306
#>
#> Group : GRADUATE
#> Class 1/3 Class 2/3
#> (Intercept) 2.6492 1.3620
#> SEXFEMALE -1.1140 -1.0306
#>
#> Item-response probabilities (Y = 1) :
#> DEFECT HLTH RAPE POOR SINGLE NOMORE
#> Class 1 1.0000 1.0000 1.0000 0.9836 0.9309 0.9682
#> Class 2 0.8342 0.9488 0.8086 0.0700 0.0440 0.1409
#> Class 3 0.0649 0.3825 0.0989 0.0000 0.0000 0.0000
#>
#> Item-response probabilities (Y = 2) :
#> DEFECT HLTH RAPE POOR SINGLE NOMORE
#> Class 1 0.0000 0.0000 0.0000 0.0164 0.0691 0.0318
#> Class 2 0.1658 0.0512 0.1914 0.9300 0.9560 0.8591
#> Class 3 0.9351 0.6175 0.9011 1.0000 1.0000 1.0000The estimated coefficients for SEX and their odds ratios
show that females are less advocate for legalizing abortion compared to
their male counterparts.
Tobacco smoking behavior
The dataset nyts18 comprises five dichotomized manifest
items on the life-time experience of several types of tobacco including
cigarettes (ECIGT), cigars (ECIGAR), chewing
tobacco/snuff/or dip (ESLT), electronic cigarettes
(EELCIGT), and hookah or water pipe (EHOOKAH)
taken from the National Youth Tobacco Survey 2018 (NYTS
2018, https://www.cdc.gov/tobacco/). The sample considered in
this study includes 1,743 non-Hispanic white students from 45 schools.
The number of sampled students from each school is in the range of 30 to
50. The school membership can be identified by SCH_ID and
each school is classified as either middle or high school
(SCH_LEV). (?nyts18)
According to socioecological models, patterns of adolescent tobacco
smoking are best understood as embedded within social contexts. These
social contexts can be either proximal in terms of individuals and peer
groups or more distal in terms of schools and community. Socioecological
models suggest that students within the same school often share common
socio-economic status (SES) and cultural characteristics that may cause
different tobacco smoking patterns compared to students attending other
schools. However, reflecting school (group) effect in an mgLCR is less
likely to provide a meaningful summary because there are too many
groups, that is, 45 schools in the nyts18 dataset. In this
case, npLCR would be more appropriate model to investigate group
difference in terms of a small number of latent clusters of schools.
data("nyts18")
str(nyts18)
#> 'data.frame': 1734 obs. of 8 variables:
#> $ ECIGT : Factor w/ 2 levels "Yes","No": 2 2 2 2 2 2 2 2 2 2 ...
#> $ ECIGAR : Factor w/ 2 levels "Yes","No": 2 1 2 2 2 1 2 2 2 2 ...
#> $ ESLT : Factor w/ 2 levels "Yes","No": 2 2 2 2 2 2 2 2 2 2 ...
#> $ EELCIGT: Factor w/ 2 levels "Yes","No": 2 1 2 2 2 1 2 2 2 2 ...
#> $ EHOOKAH: Factor w/ 2 levels "Yes","No": 2 2 2 2 2 2 2 2 2 2 ...
#> $ SEX : Factor w/ 2 levels "Male","Female": 2 1 1 1 1 2 1 2 1 2 ...
#> $ SCH_ID : Factor w/ 45 levels "d3c3dc","12d5ad",..: 11 11 11 11 11 11 11 11 11 11 ...
#> $ SCH_LEV: Factor w/ 2 levels "High School",..: 1 1 1 1 1 1 1 1 1 1 ...Selecting the number of latent classes: Prior to
conducting npLCR, it is necessary to determine the number of level-1
latent classes. Similar to the previous example, the two-, three-, and
four-class standard LCA models can be fitted and compared by the
gofglca() function as follow:
f <- item(starts.with = "E") ~ 1
lca2 <- glca(f, data = nyts18, nclass = 2, seed = 1, verbose = FALSE)
lca3 <- glca(f, data = nyts18, nclass = 3, seed = 1, verbose = FALSE)
lca4 <- glca(f, data = nyts18, nclass = 4, seed = 1, verbose = FALSE)
gofglca(lca2, lca3, lca4, test = "boot", seed = 1)
#> Model 1: item(starts.with = "E") ~ 1
#> nclass: 2
#> Model 2: item(starts.with = "E") ~ 1
#> nclass: 3
#> Model 3: item(starts.with = "E") ~ 1
#> nclass: 4
#>
#> Goodness of Fit Table :
#> logLik AIC CAIC BIC entropy Res.Df Gsq Boot p-value
#> 1 -2119.91 4261.83 4332.87 4321.87 0.92 20 96.48 0.00
#> 2 -2086.86 4207.71 4317.50 4300.50 0.87 14 30.37 0.28
#> 3 -2082.87 4211.74 4360.28 4337.28 0.82 8 22.40 0.78
#>
#> Analysis of Deviance Table :
#> npar logLik Df Deviance Boot p-value
#> 1 11 -2119.91
#> 2 17 -2086.86 6 66.11 0.0
#> 3 23 -2082.87 6 7.97 0.1The output of the gofglca() function indicates that the
three-class and the four-class models are adequate in terms of absolute
model fit (\(p\)-value = 0.28 and 0.78,
respectively), but the three-class LCA provides the lowest values in
information criteria. For the relative model fit, the two-class LCA is
rejected (\(p\)-value = 0.00) on
comparison with the three-class LCA using the bootstrap. However, the
three-class is not rejected (\(p\)-value = 0.10) on comparison with the
four-class model. Thus, it seems to be reasonable to select the
three-class LCA model for the nyts18 dataset.
Selecting the number of latent clusters: Now, an
npLCR can be implemented by adding SCH_ID as group variable and
specifying the number of latent clusters (i.e., level-2 latent classes)
using the ncluster argument in the glca()
function. The two-, three-, and four-cluster npLCR with three latent
classes are fitted and compared using the following commands:
nplca2 <- glca(f, group = SCH_ID, data = nyts18, nclass = 3, ncluster = 2, seed = 1, verbose = FALSE)
nplca3 <- glca(f, group = SCH_ID, data = nyts18, nclass = 3, ncluster = 3, seed = 1, verbose = FALSE)
nplca4 <- glca(f, group = SCH_ID, data = nyts18, nclass = 3, ncluster = 4, seed = 1, verbose = FALSE)
gofglca(lca3, nplca2, nplca3, nplca4, test = "boot", seed = 1)
#> Model 1: item(starts.with = "E") ~ 1
#> nclass: 3
#> Model 2: item(starts.with = "E") ~ 1
#> Group: SCH_ID, nclass: 3, ncluster: 2
#> Model 3: item(starts.with = "E") ~ 1
#> Group: SCH_ID, nclass: 3, ncluster: 3
#> Model 4: item(starts.with = "E") ~ 1
#> Group: SCH_ID, nclass: 3, ncluster: 4
#>
#> Goodness of Fit Table :
#> logLik AIC CAIC BIC entropy Res.Df Gsq Boot p-value
#> 1 -2086.86 4207.71 4317.50 4300.50 0.87 14 30.37 0.20
#> 2 -1955.49 3950.97 4080.14 4060.14 0.84 1419 765.73 0.08
#> 3 -1938.73 3923.46 4072.00 4049.00 0.84 1416 732.22 0.22
#> 4 -1938.30 3928.60 4096.51 4070.51 0.84 1413 731.35 0.24
#>
#> Analysis of Deviance Table :
#> npar logLik Df Deviance Boot p-value
#> 1 17 -2086.86
#> 2 20 -1955.49 3 262.74 0.00
#> 3 23 -1938.73 3 33.51 0.00
#> 4 26 -1938.30 3 0.87 0.42The goodness-of-fit table shows that the three-cluster model (Model 3) has the smallest values in information criteria among others, and the bootstrap \(p\)-value (= 0.22) indicates that this model is appropriate for the data. The analysis-of-deviance table shows that group effect is significant as Model 1 has better fit than Model 2 (\(p\)-value = 0.00). In addition, the three-cluster model (Model 3) has better fit than the two-cluster model (Model 2, \(p\)-value = 0.00), but there is insignificant difference between the three- and four-cluster models (Model 3 and Model 4) in the model fit (\(p\)-value = 0.42). Therefore, we can conclude that the three-cluster model is most appropriate among others.
Testing the equality of coefficients for level-1 covariates
across groups: Covariates can be incorporated into npLCR using
the glca() function; not only the subject-specific (i.e.,
level-1) covariates (e.g., SEX) but also the group-specific
(i.e., level-2) covariates (e.g., SCH_LEV). As shown in the
previous example, subject-specific covariates are constrained to be
equal across latent clusters by default (i.e.,
coeff.inv = TRUE). The chi-square LRT test for checking the
equality of coefficients for level-1 covariate, SEX can be
conducted using the gofglca() function as follows:
fx <- item(starts.with = "E") ~ SEX
nplcr1 <- glca(fx, group = SCH_ID, data = nyts18, nclass = 3, ncluster = 2, seed = 1, verbose = FALSE)
nplcr2 <- glca(fx, group = SCH_ID, data = nyts18, nclass = 3, ncluster = 2, seed = 1, coeff.inv = FALSE, verbose = FALSE)
nplcr.gof <- gofglca(nplca3, nplcr1, nplcr2, test = "chisq")
nplcr.gof$dtable
#> npar logLik Df Deviance Pr(>Chi)
#> 2 22 -1951.879 NA NA NA
#> 1 23 -1938.731 1 26.294 2.931509e-07
#> 3 24 -1949.498 1 -21.533 1.000000e+00Note that typing nplca.gof$dtable into the R console
will return the analysis-of-deviance table. Considering the \(p\)-values given in the table above, the
equality of SEX effect can be assumed. Therefore, we conclude that the
model specified in nplcr1 should be selected for the
nyts18 dataset.
coef(nplcr1)
#>
#> Level 1 Coefficients :
#>
#> Class 1 / 3 :
#> Odds Ratio Coefficient Std. Error t value Pr(>|t|)
#> SEXFemale 1.05946 0.05776 0.16491 0.35 0.726
#>
#> Class 2 / 3 :
#> Odds Ratio Coefficient Std. Error t value Pr(>|t|)
#> SEXFemale 1.9301 0.6576 0.2071 3.176 0.00152 **
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
plot(nplcr1)
Incorporating the level-2 covariates: In npLCR, the
meaning of the latent cluster (i.e., level-2 class) is interpreted by
the prevalence of latent class (i.e., level-1 class) for each of cluster
membership. When level-2 covariates are incorporated into the model,
this prevalence is designed to be affected by level-2 covariates as
shown (5). In other words, as level-2 covariates significantly influence
the prevalence of level-1 class, the meaning of latent cluster may
change as the level of level-2 covariate changes. Therefore, the number
of clusters could be reduced when level-2 covariates are included
compared to the number of clusters of npLCR without any level-2
covariate. As we selected the three-cluster model without any level-2
covariate for the nyts18 dataset, the fit of the
three-cluster model should be compared with the fit of the two-cluster
model when SCH_LEV is incorporated as a level-2 covariate. Note that
different seed values (seed = 3 and seed = 6)
are used in the glca() function below in order to produce
the class order displayed in Figure 2.
f.2 <- item(starts.with = "E") ~ SEX + SCH_LEV
nplcr3 <- glca(f.2, group = SCH_ID, data = nyts18, nclass = 3, ncluster = 2, seed = 3, verbose = FALSE)
nplcr4 <- glca(f.2, group = SCH_ID, data = nyts18, nclass = 3, ncluster = 3, seed = 6, verbose = FALSE)
gofglca(nplcr3, nplcr4, test = "boot", seed = 1)
#> Model 1: item(starts.with = "E") ~ SEX + SCH_LEV
#> Group: SCH_ID, nclass: 3, ncluster: 2, coeff.inv: TRUE
#> Model 2: item(starts.with = "E") ~ SEX + SCH_LEV
#> Group: SCH_ID, nclass: 3, ncluster: 3, coeff.inv: TRUE
#>
#> Goodness of Fit Table :
#> logLik AIC CAIC BIC entropy Res.Df Gsq Boot p-value
#> 1 -1919.94 3887.87 4042.87 4018.87 0.83 1709 1052.54 0.12
#> 2 -1916.30 3886.60 4060.97 4033.97 0.84 1706 1045.26 0.06
#>
#> Analysis of Deviance Table :
#> npar logLik Df Deviance Boot p-value
#> 1 24 -1919.94
#> 2 27 -1916.30 3 7.28 0The analysis-of-deviance table from the gofglca() function shows that
three clusters are required even when a level-2 covariate SCH_LEV is
incorporated (\(p\)-value = 0.00).
However, information criteria and \(p\)-values for absolute model fit may let
us reach a different conclusion: the two-cluster model provides smaller
values in some information criteria and the \(p\)-value (= 0.12) indicates that the model
is appropriate for the dataset. Without strong prior beliefs, the number
of latent clusters should be chosen to strike a balance between
parsimony, fit, and interpretability. Two bar graphs in Figure 2 display
the estimated class prevalences for each cluster membership from the
models specified in nplcr3 and nplcr4 using
the plot() function, respectively. The values in
parentheses printed in the x-axis are the estimated cluster prevalence.
The bar graph in the right, which is generated from the object
nplcr4, shows that there is not much difference in class
prevalence among these three clusters, compared to the bar graph in the
left generated from the two-cluster model specified in
nplcr3. In other words, the three-cluster model is not
substantively meaningful, and therefore, we may conclude that the
two-cluster model specified in nplcr3 is more adequate to
describe cluster (group) variation in the latent class distribution.
Summarizing the results from the selected model: The
estimated parameters from the three-class and two-cluster npLCR model
specified in nplcr3 are shown in Figure 3 using the
plot() function. Based on the line graph in Figure 3, we
deduce that Class 1 represents the poly-user group; Class 2 is the
electronic cigarette user group; and Class 3 is the non-smoking group.
We already argued that the two latent clusters were clearly
distinguished by their class prevalence using the left bar graph in
Figure 2. According to this stacked bar graph, the probability of
engagement in a certain latent class is significantly different between
these two school clusters. About 12% and 29% of students in Cluster 2
belong to Class 1 (poly-user group) and Class 2 (electronic cigarette
user group), whereas only 2% and 9% of them belong to Class 1 and Class
2, respectively. The difference in class prevalences by cluster
indicates that students who attend a school classified as Cluster 2 are
more likely to be smokers relative to a similar student attending a
school classified as Cluster 1. The full output of the
glca() function for the selected model can be displayed by
the summary() function as follows:
summary(nplcr3)
#>
#> Call:
#> glca(formula = f.2, group = SCH_ID, data = nyts18, nclass = 3,
#> ncluster = 2, seed = 3, verbose = FALSE)
#>
#> Manifest items : ECIGAR ECIGT EELCIGT EHOOKAH ESLT
#> Grouping variable : SCH_ID
#> Covariates (Level 1) : SEX
#> Covariates (Level 2) : SCH_LEV
#>
#> Categories for manifest items :
#> Y = 1 Y = 2
#> ECIGAR Yes No
#> ECIGT Yes No
#> EELCIGT Yes No
#> EHOOKAH Yes No
#> ESLT Yes No
#>
#> Model : Nonparametric multilevel latent class analysis
#>
#> Number of latent classes : 3
#> Number of latent clusters : 2
#> Number of groups : 45
#> Number of observations : 1734
#> Number of parameters : 24
#>
#> log-likelihood : -1919.937
#> G-squared : 1052.541
#> AIC : 3887.874
#> BIC : 4018.87
#>
#> Marginal prevalences for latent classes :
#> Class 1 Class 2 Class 3
#> 0.06455 0.18159 0.75385
#>
#> Marginal prevalences for latent clusters :
#> Cluster 1 Cluster 2
#> 0.54083 0.45917
#>
#> Class prevalences by cluster :
#> Class 1 Class 2 Class 3
#> Cluster 1 0.0199 0.08978 0.89032
#> Cluster 2 0.1175 0.29044 0.59206
#>
#> Logistic regression coefficients (level 1) :
#> Cluster 1
#> Class 1/3 Class 2/3
#> (Intercept) -2.6345 -1.0654
#> SEXFemale 0.6307 0.1159
#>
#> Cluster 2
#> Class 1/3 Class 2/3
#> (Intercept) -0.8997 0.1671
#> SEXFemale 0.6307 0.1159
#>
#> Logistic regression coefficients (level 2) :
#> Class 1/3 Class 2/3
#> SCH_LEVMiddle School -2.5656 -1.9522
#>
#> Item-response probabilities (Y = 1) :
#> ECIGAR ECIGT EELCIGT EHOOKAH ESLT
#> Class 1 0.9657 0.8914 0.9782 0.5049 0.5507
#> Class 2 0.1696 0.3227 0.7266 0.0394 0.1127
#> Class 3 0.0034 0.0033 0.0372 0.0056 0.0074
#>
#> Item-response probabilities (Y = 2) :
#> ECIGAR ECIGT EELCIGT EHOOKAH ESLT
#> Class 1 0.0343 0.1086 0.0218 0.4951 0.4493
#> Class 2 0.8304 0.6773 0.2734 0.9606 0.8873
#> Class 3 0.9966 0.9967 0.9628 0.9944 0.9926As shown in the previous example, the result of Wald test for each of
estimated odds ratios and coefficients can be obtained by typing
coef(nplcr3) into the R console (results not shown here).
The Wald test shows that females are more likely to belong to Class 1
than Class 3, indicating that females are at a higher risk than their
male counterparts. For level-2 covariate SCH_LEV, middle
schools are less likely to belong to Class 1 or 2 than Class 3, that is,
middle-school students tend to smoke less than high-school students.
Imputing the cluster membership: Researchers often
want to explore the effects of level-2 covariates on the imputed latent
cluster membership. Note that we selected the two-cluster model
specified in nplcr3. We can easily impute the latent
cluster membership for 45 schools using the posterior probabilities from
the model specified in nplcr3 and fit the standard logistic
regression with SCH_LEV as a covariate. The posterior
probabilities for latent cluster can be accessed by
nplcr3$posterior$cluster. The following codes generate the
imputed latent cluster membership for each school and save the cluster
membership in ndata with level-2 covariate
SCH_LEV.
tmp1 <- unique(nyts18[c("SCH_ID", "SCH_LEV")])
tmp2 <- nplcr3$posterior$cluster
tmp3 <- data.frame(SCH_ID = rownames(tmp2), Cluster = factor(apply(tmp2, 1, which.max)))
ndata <- merge(tmp1, tmp3)
head(ndata)
#> SCH_ID SCH_LEV Cluster
#> 1 00b895 Middle School 1
#> 2 066e6c High School 2
#> 3 0690d1 High School 2
#> 4 0fc94b High School 2
#> 5 12d5ad Middle School 1
#> 6 16082c Middle School 1Using the logistic regression, the effect of level-2 covariate
SCH_LEV on the latent cluster membership can be estimated
as following:
fit <- glm(Cluster ~ SCH_LEV, family = binomial, data = ndata)
summary(fit)
#>
#> Call:
#> glm(formula = Cluster ~ SCH_LEV, family = binomial, data = ndata)
#>
#> Deviance Residuals:
#> Min 1Q Median 3Q Max
#> -1.4823 -0.9005 -0.9005 0.9005 1.4823
#>
#> Coefficients:
#> Estimate Std. Error z value Pr(>|z|)
#> (Intercept) 0.6931 0.5477 1.266 0.2057
#> SCH_LEVMiddle School -1.3863 0.6708 -2.067 0.0388 *
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> (Dispersion parameter for binomial family taken to be 1)
#>
#> Null deviance: 61.827 on 44 degrees of freedom
#> Residual deviance: 57.286 on 43 degrees of freedom
#> AIC: 61.286
#>
#> Number of Fisher Scoring iterations: 4